11.9.4 Vertical Composition of Natural Transformations

    Definition 11.9.4.1.1 ▶ Vertical Composition of Natural Transformations

    The vertical composition of two natural transformations $\alpha \colon F\Longrightarrow G$ and $\beta \colon G\Longrightarrow H$ as in the diagram

    is the natural transformation $\beta \circ \alpha \colon F\Longrightarrow H$ consisting of the collection

    \[ \left\{ (\beta \circ \alpha )_{A} \colon F(A) \to H(A) \right\} _{A\in \operatorname {\mathrm{Obj}}(\mathcal{C})} \]

    with

    \[ (\beta \circ \alpha )_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\beta _{A}\circ \alpha _{A} \]

    for each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$.

    Proof of Definition 11.9.4.1.1.

    The naturality condition for $\beta \circ \alpha $ is the requirement that the boundary of the diagram

    commutes. Since

    • •

      Subdiagram (1) commutes by the naturality of $\alpha $.

    • •

      Subdiagram (2) commutes by the naturality of $\beta $.

    so does the boundary diagram. Hence $\beta \circ \alpha $ is a natural transformation.

    Proposition 11.9.4.1.2 ▶ Properties of Vertical Composition of Natural Transformations

    Let $\mathcal{C}$, $\mathcal{D}$, and $\mathcal{E}$ be categories.

    1. 1.

      Functionality. The assignment $(\beta ,\alpha )\mapsto \beta \circ \alpha $ defines a function

      \[ \circ _{F,G,H}\colon \operatorname {\mathrm{Nat}}(G,H)\times \operatorname {\mathrm{Nat}}(F,G)\to \operatorname {\mathrm{Nat}}(F,H). \]
    2. 2.

      Associativity. Let $F,G,H,K\colon \mathcal{C}\overset {\rightrightarrows }{\rightrightarrows }\mathcal{D}$ be functors. The diagram

      commutes, i.e. given natural transformations

      \[ F\mathbin {\overset {\alpha }{\Longrightarrow }}G\mathbin {\overset {\beta }{\Longrightarrow }}H\mathbin {\overset {\gamma }{\Longrightarrow }}K, \]

      we have

      \[ (\gamma \circ \beta )\circ \alpha =\gamma \circ (\beta \circ \alpha ). \]
  • 3.

    Unitality. Let $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors.

    1. (a)

      Left Unitality. The diagram

      commutes, i.e. given a natural transformation $\alpha \colon F\Longrightarrow G$, we have

      \[ \operatorname {\mathrm{id}}_{G}\circ \alpha =\alpha . \]
    2. (b)

      Right Unitality. The diagram

      commutes, i.e. given a natural transformation $\alpha \colon F\Longrightarrow G$, we have

      \[ \alpha \circ \operatorname {\mathrm{id}}_{F}=\alpha . \]
  • 4.

    Middle Four Exchange. Let $F_{1},F_{2},F_{3}\colon \mathcal{C}\to \mathcal{D}$ and $G_{1},G_{2},G_{3}\colon \mathcal{D}\to \mathcal{E}$ be functors. The diagram

    commutes, i.e. given a diagram
    in $\mathsf{Cats}_{\mathsf{2}}$, we have

    \[ (\beta '\mathbin {\star }\alpha ')\circ (\beta \mathbin {\star }\alpha )=(\beta '\circ \beta )\mathbin {\star }(\alpha '\circ \alpha ). \]
  • 5.

    Interaction With Natural Isomorphisms. If $\alpha $ and $\beta $ are natural isomorphisms, then so is $\beta \circ \alpha $.

    • Proof of Proposition 11.9.4.1.2.

    Item 1: Functionality
    This follows from the well-definedness of Definition 11.9.4.1.1.

    Item 2: Associativity
    Indeed, we have

    \begin{align*} ((\gamma \circ \beta )\circ \alpha )_{A} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(\gamma \circ \beta )_{A}\circ \alpha _{A}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(\gamma _{A}\circ \beta _{A})\circ \alpha _{A}\\ & = \gamma _{A}\circ (\beta _{A}\circ \alpha _{A})\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\gamma _{A}\circ (\beta \circ \alpha )_{A}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(\gamma \circ (\beta \circ \alpha ))_{A} \end{align*}

    for each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, showing the desired equality.

    Item 3: Unitality
    We have

    \begin{align*} (\operatorname {\mathrm{id}}_{G}\circ \alpha )_{A} & = \operatorname {\mathrm{id}}_{G}\circ \alpha _{A}\\ & = \alpha _{A},\\ (\alpha \circ \operatorname {\mathrm{id}}_{F})_{A} & = \alpha _{A}\circ \operatorname {\mathrm{id}}_{F}\\ & = \alpha _{A} \end{align*}

    for each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, showing the desired equality.

    Item 4: Middle Four Exchange
    This is proved in Item 4 of Proposition 11.9.5.1.3.

    Item 5: Interaction With Natural Isomorphisms
    Omitted.

Up

View Item 3 as PDF
1 tag refers to this tag
Statistics Cite

Noticed something off, or have any comments? Feel free to reach out!

You can also use the contact form below: